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<span class="kw">let </span><font color="Maroon" title="c1">D</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">I</font>, <font color="Olive" title="b2">J</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b3">F</font>, <font color="Olive" title="b4">G</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">D</font><br/> for <font color="Olive" title="b5">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b1">I</font>,<font color="Maroon" title="c1">D</font><br/> for <font color="Olive" title="b6">g</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b2">J</font>,<font color="Maroon" title="c1">D</font>  st <font color="Olive" title="b3">F</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  &amp; <font color="Olive" title="b3">F</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  holds <br/>for <font color="Olive" title="b7">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">I</font><br/> for <font color="Olive" title="b8">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b2">J</font> holds  <font color="Olive" title="b3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b7">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b8">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<span class="p1">(<span class="default"><font color="Olive" title="b4">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Olive" title="b5">f</font>,<font color="Olive" title="b6">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b7">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<span class="p1">(<span class="default"><font color="Olive" title="b4">G</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Olive" title="b5">f</font>,<span class="p2">(<span class="default"><font color="Olive" title="b3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p3"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b8">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<font color="Olive" title="b6">g</font></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">I</font>, <font color="Maroon" title="c3">J</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">F</font>, <font color="Olive" title="b2">G</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">D</font><br/> for <font color="Olive" title="b3">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c1">D</font><br/> for <font color="Olive" title="b4">g</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">J</font>,<font color="Maroon" title="c1">D</font>  st <font color="Olive" title="b1">F</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  &amp; <font color="Olive" title="b1">F</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  holds <br/>for <font color="Olive" title="b5">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font><br/> for <font color="Olive" title="b6">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">J</font> holds  <font color="Olive" title="b1">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b5">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b6">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Olive" title="b3">f</font>,<font color="Olive" title="b4">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b5">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">G</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Olive" title="b3">f</font>,<span class="p2">(<span class="default"><font color="Olive" title="b1">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p3"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b6">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<font color="Olive" title="b4">g</font></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">F</font>, <font color="Maroon" title="c5">G</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c1">D</font><br/> for <font color="Olive" title="b2">g</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">J</font>,<font color="Maroon" title="c1">D</font>  st <font color="Maroon" title="c4">F</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  &amp; <font color="Maroon" title="c4">F</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  holds <br/>for <font color="Olive" title="b3">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font><br/> for <font color="Olive" title="b4">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">J</font> holds  <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b3">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b4">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b3">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Olive" title="b1">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p3"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b4">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<font color="Olive" title="b2">g</font></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c1">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">g</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">J</font>,<font color="Maroon" title="c1">D</font>  st <font color="Maroon" title="c4">F</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  &amp; <font color="Maroon" title="c4">F</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  holds <br/>for <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font><br/> for <font color="Olive" title="b3">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">J</font> holds  <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b2">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b3">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Olive" title="b1">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b2">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c6">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p3"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b3">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<font color="Olive" title="b1">g</font></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c7">g</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">J</font>,<font color="Maroon" title="c1">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">F</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  &amp; <font color="Maroon" title="c4">F</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  implies for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font><br/> for <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">J</font> holds  <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b2">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c6">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p3"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b2">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<font color="Maroon" title="c7">g</font></span>)</span></span>)</span> )</span><br/>









<span class="kw">assume </span><a NAME="E1:36"/><span class="lab"><font color="Green" title="E15">A1</font></span>: 
( <font color="Maroon" title="c4">F</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  &amp; <font color="Maroon" title="c4">F</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font><br/> for <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">J</font> holds  <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b2">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c6">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p3"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b2">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<font color="Maroon" title="c7">g</font></span>)</span></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">G</font> = <font color="Maroon" title="c5">G</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c1">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c1">D</font> ;<br/>
<a NAME="E3:36"/><span class="lab"><font color="Green" title="E16">A2</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">J</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">J</font> holds  <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b1">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c6">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p3"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b1">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<font color="Maroon" title="c7">g</font></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c10">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">J</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c10">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c6">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p3"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c10">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<font color="Maroon" title="c7">g</font></span>)</span></span>)</span></span><br/><a NAME="E1:36_1"/><span class="lab"><font color="Green" title="E17">A3</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">J</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">z</font> = <font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">u</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">J</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><span class="lab"><font color="Green" title="E18">A4</font></span>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">f</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c11">z</font></span>)</span></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c11">z</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D8" title="FUNCT_3:def.8">FUNCT_3:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></span>
<br/>.= 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">f</font>

;<br/><a NAME="E5:36_1"/>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c11">z</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/><a NAME="E6:36_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A5</font></span>: 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c11">z</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c1">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></span>;<br/><a NAME="E7:36_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">I</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E8:36_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A6</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">f</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c11">z</font></span>)</span></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:36_1"><span class="lab"><font color="Green" title="E18">A4</font></span></a></span>;<br/><a NAME="E9:36_1"/>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c1">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">f</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c11">z</font></span>)</span></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c11">z</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="funct_3.html#T71" title="FUNCT_3:th.71">FUNCT_3:71</a></span>;<br/><a NAME="E10:36_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">f</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c11">z</font></span>)</span></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:36_1"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="txt" href="#E6:36_1"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E11:36_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A7</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c11">z</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#D4" title="FUNCOP_1:def.4">FUNCOP_1:def 4</a></span>;<br/><span class="lab"><font color="Green" title="E22">A8</font></span>: <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c6">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p3"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c10">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<font color="Maroon" title="c7">g</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c6">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p3"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c10">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<font color="Maroon" title="c7">g</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="setwiseo.html#T26" title="SETWISEO:th.26">SETWISEO:26</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c6">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="setwiseo.html#T26" title="SETWISEO:th.26">SETWISEO:26</a></span>
<br/>.= 
<font color="Maroon" title="c8">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:36_1"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="ref" href="funcop_1.html#T35" title="FUNCOP_1:th.35">FUNCOP_1:35</a></span>
;<br/><font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c10">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> = 
<font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c12">u</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T35" title="ZFMISC_1:th.35">ZFMISC_1:35</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="setwiseo.html#T26" title="SETWISEO:th.26">SETWISEO:26</a></span>
<br/>.= 
<font color="Maroon" title="c8">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="txt" href="#E1:36_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="ref" href="finseqop.html#T82" title="FINSEQOP:th.82">FINSEQOP:82</a></span>
;<br/><span class="kw">hence </span><a NAME="E14:36_1"/>
<font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c10">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c6">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p3"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c10">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<font color="Maroon" title="c7">g</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:36_1"><span class="lab"><font color="Green" title="E22">A8</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:36"/>
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font><br/> for <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">J</font> holds  <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b2">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c6">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p3"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b2">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<font color="Maroon" title="c7">g</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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